Prime numbers and random walks in a square grid.

In recent years, computer simulations have played a fundamental role in unveiling some of the most intriguing features of prime numbers. In this paper, we define an algorithm for a deterministic walk through a two-dimensional grid, which we refer to as a prime walk. The walk is constructed from a sequence of steps dictated by and dependent on the sequence of the last digits of the primes. Despite the apparent randomness of this generating sequence, the resulting structure-in both two and three dimensions-created by the algorithm presents remarkable properties and regularities in its pattern, which we proceed to analyze in detail.

New effective method for quantitative analysis of diffusion jumps, applied in molecular dynamics simulations of small molecules dispersed in short chain systems.

Diffusion jumps of small molecules dispersed in chain molecules or other kinds of slow-moving matrices have already been observed in many previous simulations of such systems, and their treatment led to important qualitative conclusions. In the present work, a new, very simple yet effective method is described, allowing for both identification of individual penetrant jump events and their quantitative treatment in a statistical sense. The method is applied in equilibrium Molecular Dynamics simulations for systems of gaseous alkanes, methane through n-butane, including also a mixture of methane and n-butane, dispersed in n-decane or n-eicosane. Equilibration and attainment of a linear diffusion regime is confirmed by means of various criteria, and the jumps detection method is applied to all systems studied. The results obtained clearly show the existence of distinct jump events in all cases, although the average jump length is reduced with penetrant or liquid alkane molecular weight. The method allows one to determine the average jump length and the corresponding jumps frequency. On the basis of these results, it was possible to estimate a random walk type diffusion coefficient, D(s,jumps), of the penetrants, which was found to be substantially lower compared with the overall diffusion coefficient D(s,MSD) obtained by the mean square displacement method. This finding led us to assume that the overall penetrants' diffusion in the studied systems is a combination of longer jumps with a smoother and more gradual displacement, a result that confirms assumptions suggested in previous studies.